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CHAPTER 7

INTRODUCTION TO CLASSICAL SETS AND

FUZZY SETS

FUZZY LOGICFUZZY LOGIC Fuzzy logic is the logic underlying approximate, rather than

exact, modes of reasoning.

It is an extension of multivalued logic: Everything, including truth, is a matter of degree.

It contains as special cases not only the classical two-value logic and multivalue logic systems, but also probabilistic logic.

A proposition p has a truth value• 0 or 1 in two-value system,• element of a set T in multivalue system,• Range over the fuzzy subsets of T in fuzzy logic.

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FUZZY LOGICFUZZY LOGIC

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Fuzzy logic is a set of mathematical principles forknowledge representation and reasoning based on degrees of membership.

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TYPES AND MODELING OF UNCERTAINTY

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FUZZY vs PROBABILITY

Fuzzy ≠ Probability

Probability deals with uncertainty an likelihood

Fuzzy logic deals with ambiguity an vagueness

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FUZZY vs PROBABILITY

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NEED OF FUZZY LOGIC Based on intuition and judgment.

No need for a mathematical model.

Provides a smooth transition between members and nonmembers.

Relatively simple, fast and adaptive.

Less sensitive to system fluctuations.

Can implement design objectives, difficult to express mathematically, in linguistic or descriptive rules.

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CLASSICAL SETS (CRISP SETS)

Conventional or crisp sets are Binary. An element either belongs to the set or does not.

{True, False}

{1, 0}

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CRISP SETS

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OPERATIONS ON CRISP SETS

UNION:

INTERSECTION:

COMPLEMENT:

DIFFERENCE:

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PROPERTIES OF CRISP SETSThe various properties of crisp sets are as follows:

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PROPERTIES OF CRISP SETS

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FUZZY SETS

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FUZZY SETS

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FUZZY SETS

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OPERATIONS ON FUZZY SETS

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PROPERTIES OF FUZZY SETS

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PROPERTIES OF FUZZY SETS

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PROPERTIES OF FUZZY SETS

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RELATIONS Relations represent mappings between sets and connectives in logic.

A classical binary relation represents the presence or absence of a connection or interaction or association between the elements of two sets.

Fuzzy binary relations are a generalization of crisp binary relations, and they allow various degrees of relationship (association) between elements.

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CRISP CARTESIAN PRODUCT

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CRISP RELATIONS

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CRISP RELATIONS

CRISP BINARY CRISP BINARY RELATIONSRELATIONS

Examples of binary relations

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OPERATIONS ON CRISP RELATIONS

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PROPERTIES OF CRISP RELATIONSThe properties of crisp sets (given below)

hold good for crisp relations as well.

Commutativity,

Associativity,

Distributivity,

Involution,

Idempotency,

DeMorgan’s Law,

Excluded Middle Laws.

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COMPOSITION ON CRISP RELATIONS

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FUZZY CARTESIAN PRODUCTLet R be a fuzzy subset of M and S be a fuzzy subset of N.

Then the Cartesian product R × S is a fuzzy subset of N × M such that

Example:Let R be a fuzzy subset of {a, b, c} such that R = a/1 + b/0.8 + c/0.2 and S be a fuzzy subset of {1, 2, 3} such that S = 1/1 + 3/0.8 + 2/0.5. Then R x S is given by

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FUZZY RELATION

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FUZZY RELATION

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OPERATIONS ON FUZZY RELATIONThe basic operation on fuzzy sets also apply on fuzzy

relations.

OPERATIONS ON FUZZY OPERATIONS ON FUZZY RELATIONRELATION

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PROPERTIES OF FUZZY RELATIONSThe properties of fuzzy sets (given below)

hold good for fuzzy relations as well.

Commutativity,

Associativity,

Distributivity,

Involution,

Idempotency,

DeMorgan’s Law,

Excluded Middle Laws.

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COMPOSITION OF FUZZY RELATIONS

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COMPOSITION OF FUZZY RELATIONS

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COMPOSITION OF FUZZY RELATIONS

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COMPOSITION OF FUZZY RELATIONS

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CLASSICAL EQUIVALENCE RELATION

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CLASSICAL TOLERANCE RELATION

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FUZZY EQUIVALENCE RELATION

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FUZZY TOLERANCE RELATIONA binary fuzzy relation that possesses the properties of

reflexivity and symmetry is called fuzzy tolerance relation or resemblance relation.

The equivalence relations are a special case of the tolerance relation. The fuzzy tolerance relation can be reformed into fuzzy equivalence relation in the same way as a crisp tolerance relation is reformed into crisp equivalence relation, i.e.,

where ‘n’ is the cardinality of the set that defines R1.

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CRISP MEMBERSHIP FUCNTIONS

Crisp membership functions (µ ) are either one or zero.

Consider the example: Numbers greater than 10. The membership curve for the set A is given by

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REPRESENTING A DOMAIN IN FUZZY

LOGIC

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FUZZY MEMBERSHIP FUCNTIONS

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FUZZY MEMBERSHIP FUCNTIONS

The set B of numbers approaching 2 can be represented by the membership function

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FUZZINESS vs PROBABILITY

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LINGUISTIC VARIABLE

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LINGUISTIC VARIABLE Let x be a linguistic variable with the label “speed”. Terms of x, which are fuzzy sets, could be “positive low”, “negative high” from the term set T:

T = {PostiveHigh, PositiveLow, NegativeLow, NegativeHigh, Zero}

Each term is a fuzzy variable defined on the base variable which might be the scale of all relevant velocities.

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MEMBERSHIP FUCNTIONS

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MEMBERSHIP FUCNTIONS

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FEATURES OF MEMBERSHIP FUNCTIONS

CORE:

SUPPORT:

BOUNDARY:

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FUZZIFICATION

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FUZZIFICATION Use crisp inputs from the user.

Determine membership values for all the relevant classes (i.e., in right Universe of Discourse).

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EXAMPLE - FUZZIFICATION

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FUZZIFICATION OF HEIGHT

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FUZZIFICATION OF WEIGHT

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METHODS OF MEMBERSHIP VALUE

ASSIGNMENTThe various methods of assigning membership values are:

Intuition,

Inference,

Rank ordering,

Angular fuzzy sets,

Neural networks,

Genetic algorithm,

Inductive reasoning.

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DEFUZZIFICATION

Defuzzification is a mapping process from a space of fuzzy control actions defined over an output universe of discourse into a space of crisp (nonfuzzy) control actions.

Defuzzification is a process of converting output fuzzy variable into a unique number .

Defuzzification process has the capability to reduce a fuzzy set into a crisp single-valued quantity or into a crisp set; to convert a fuzzy matrix into a crisp matrix; or to convert a fuzzy number into a crisp number.

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LAMBDA CUT FOR FUZZY SETS

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LAMBDA CUT FOR FUZZY RELATIONS

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METHODS OF DEFUZZIFICATIONDefuzzification is the process of conversion of a fuzzy quantity

into a precise quantity. Defuzzification methods include:

Max-membership principle,

Centroid method,

Weighted average method,

Mean-max membership,

Center of sums,

Center of largest area,

First of maxima, last of maxima.

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FUZZY DECISION

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MAX MEMBERSHIP METHOD Fuzzy set with the largest membership value is selected.

Fuzzy decision: Fn = {P, F. G, VG, E} Fn = {0.6, 0.4, 0.2, 0.2, 0}

Final decision (FD) = Poor Student If two decisions have same membership max, use the average of the two.

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CENTROID METHOD

This method is also known as center-of-mass, center-of-area, or center-of-gravity method. It is the most commonly used defuzzification method. The defuzzified output x* is defined as

where the symbol ∫ denotes an algebraic integration.

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WEIGHTED AVERAGE METHOD

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MEAN MAX MEMBERSHIP METHOD

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CENTER OF SUMS

This method employs the algebraic sum of the individual fuzzy subsets instead of their unions. The calculations here are very fast but the main drawback is that the intersecting areas are added twice. The defuzzified value x* is given by

CENTER OF LARGEST AREACENTER OF LARGEST AREA

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This method can be adopted when the output consists of at least two convex fuzzy subsets which are not overlapping. The output in this case is biased towards a side of one membership function. When output fuzzy set has at least two convex regions then the center-of-gravity of the convex fuzzy subregion having the largest area is used to obtain the defuzzified value x*. This value is given by

where is the convex subregion that has the largest area making up

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FIRST OF MAXIMA (LAST OF MAXIMA)The steps used for obtaining crisp values are as follows:

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SUMMARYDefuzzification process is essential because some engineering applications need exact values for performing the operation.

Example: If speed of a motor has to be varied, we cannot instruct to raise it “slightly”, “high”, etc., using linguistic variables; rather, it should be specified as raise it by 200 rpm or so, i.e., a specific amount of raise should be mentioned.

The method of defuzzification should be assessed on the basis of the output in the context of data available.

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FUZZY ARITHMETIC

Distributions do not have to be precise.

Requires no assumption about correlations.

Fuzzy measures are upper bounds on probability.

Fuzzy arithmetic might be a conservative way to do risk assessments.

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FUZZY NUMBERS AND THEIR ARITHMETIC

Fuzzy numbers• Fuzzy sets of the real line,• Unimodal,• Reach possibility level one.

Fuzzy arithmetic• Interval arithmetic at each possibility level.

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FEATURES OF FUZZY ARITHMETIC Fully developed arithmetic and logic

• Addition, subtraction, multiplication, division, min, max;

• Log, exp, sqrt, abs, powers, and, or, not;• Backcalculation, updating, mixtures, etc.

Very fast calculation and convenient software. Very easy to explain.

Distributional answers (not just worst case). Results robust to choice about shape.

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TYPES OF NUMBERS Scalars are well-known or mathematically

defined integers and real numbers.

Intervals are numbers whose values are not know with certainty but about which bounds can be established.

Fuzzy numbers are uncertain numbers for which, in addition to knowing a range of possible values, one can say that some values are more plausible than others.

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FUZZY NUMBERS

Fuzzy set that is unimodal and reaches 1.

Nested stack of intervals.

1

0.5

0

no YESno

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LEVEL-WISE INTERVAL ARITHMETIC

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FUZZY INTERVALS BY TRAPEZOIDAL SHAPES

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FUZZY ADDITION

0

1

0

A A+BB

0

0.5

2 4 6 8

Subtraction, multiplication, division, minimum, maximum, exponentiation, logarithms, etc. are also defined.

If distributions are multimodal, possibility theory (rather than just simple fuzzy arithmetic) is required.

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FUZZY ARITHMETIC

Interval Arithmetic[a, b] + [d, e] = [a+d, b+e][a, b] - [d, e] = [a-e, b-d][a, b] x [d, e] = [min(a.d, a.e, b.d, b.e), max(a.d, a.e, b.d, b.e)][a, b]/[d, e] = [min(a/d, a/e, b/d, b/e), max(a/d, a/e, b/d, b/e)]

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FUZZY ARITHMETIC

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FUZZY ARITHMETIC

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EXTENSION PRINCIPLE

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VARIOUS FUZZY MEASURES

Belief and Plausibility Measure.

Probability Measure.

Possibility and Necessity Measure.

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POSSIBILITY MEASURE

No single definition.

Many definitions could be used:• Subjective assessments.• Social consensus. • Measurement error.• Upper betting rates (Giles)• Extra-observational ranges (Gaines).

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FUZZY SET AS POSSIBILITY MEASURE

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GETTING FUZZY INPUTS Subjective assignments:• Make them up from highest, lowest and

best-guess estimates.

Objective consensus:• Stack up consistent interval estimates or

bridge inconsistent ones.

Measurement error:• Infer from measurement protocols.

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SUBJECTIVE ASSIGNMENTS

Triangular fuzzy numbers, e.g. [1,2,3].

Trapezoidal fuzzy numbers, e.g. [1,2,3,4].

0 1 2 3 40

0.5

1

0 1 2 3 4 50

0.5

1

Possibility

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OBJECTIVE CONSENSUS

[1000, 3000] [2000, 2400] [500, 2500] [800, 4000] [1900, 2300]

Possibility

0 2000 40000

0.5

1

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MEASUREMENT ERROR

46.8 ± 0.3 [46.5, 46.8, 47.1]

[12.32] [12.315, 12.32, 12.325]

46.5 46.7 46.9 47.10

1

Possibility

12.31 12.32 12.330

1

Possibility

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SHAPE OF X AFFECTING aX+b

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ROBUSTNESS OF THE ANSWERDifferent choices for the fuzzy number X all

yield very similar distributions for aX + b

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PROBABILITY MEASURE

A fuzzy number F is said to “enclose” a probability distribution P if,

• the left side of F is larger than P(x) for each x,• the right side of F is larger than 1-P(x) for each x.

For every event X < x and x < X, possibility is larger than the probability, so it is an upper bound.

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POSSIBILITY MEASURE

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ADVANTAGES OF FUZZY ARITHMETIC

Requires little data.

Applicable to all kinds of uncertainty.

Fully comprehensive.

Fast and easy to compute.

Doesn’t require information about correlations.

Conservative, but not hyper-conservative

In between worst case and probability.

Backcalculations easy to solve.

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LIMITATIONS OF FUZZY ARITHMETIC

Controversial. Are alpha levels comparable for different

variables? Not optimal when there're a lot of data. Can’t use knowledge of correlations to tighten

answers. Not conservative against all possible

dependencies. Repeated variables make calculations

cumbersome.

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SOFTWARES FOR FUZZY ARITHMETIC

FuziCalc • (Windows 3.1) FuziWare, 800-472-6183

Fuzzy Arithmetic C++ Library• (C code) anonymous ftp to mathct.dipmat.unict.it

and get \fuzzy\fznum*.*

Cosmet (Phaser)• (DOS, soon for Windows) acooper@sandia.gov

Risk Calc • (Windows) 800-735-4350; www.ramas.com

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SCREEN SHOT OF FUZZY CALCULATOR - FzCalc

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SUMMARY

This chapter has given an overview on Fuzzy Arithmetic and Fuzzy Measures.

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FUZZY RULES AND REASONING

The degree of an element in a fuzzyset corresponds to the truth value of aproposition in fuzzy logic systems.

LINGUISTIC VARIABLESLINGUISTIC VARIABLES

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A linguistic variable is a fuzzy variable.

• The linguistic variable speed ranges between 0 and 300 km/h and includes the fuzzy sets slow, very slow, fast, …• Fuzzy sets define the linguistic values.

Hedges are qualifiers of a linguistic variable.

• All purpose: very, quite, extremely• Probability: likely, unlikely• Quantifiers: most, several, few• Possibilities: almost impossible, quite possible

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LINGUISTIC HEDGES (LINGUISTIC

QUANTIFIERS) Hedges modify the shape of a fuzzy set.

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TRUTH TABLESTruth tables define logic functions of two propositions. Let X and Y be two propositions, either of which can be true or false.

The operations over the propositions are:

1.Conjunction (∧): X AND Y.

2.Disjunction (∨): X OR Y.

3.Implication or conditional (⇒): IF X THEN Y.

4.Bidirectional or equivalence (⇔): X IF AND ONLY IF Y.

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FUZZY RULES

A fuzzy rule is defined as the conditional statement of the form

If x is ATHEN y is B

where x and y are linguistic variables and A and B are linguistic values determined by fuzzy sets on the universes of discourse X and Y.

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FUZZY RULES The decision-making process is based on rules with sentence conjunctives AND, OR and ALSO.

Each rule corresponds to a fuzzy relation.

Rules belong to a rule base.

Example: If (Distance x to second car is SMALL) OR

(Distance y to obstacle is CLOSE) AND (speed v is HIGH) THEN (perform LARGE correction to steering angle θ ) ALSO (make MEDIUM reduction in speed v).

Three antecedents (or premises) in this example give rise to two outputs (consequences).

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FUZZY RULE FORMATION

IF height is tallTHEN weight is heavy.

Here the fuzzy classes height and weight have a given range (i.e., the universe of discourse).

range (height) = [140, 220]range (weight) = [50, 250]

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FORMATION OF FUZZY RULES

Three general forms are adopted for forming fuzzy rules. They are:

Assignment statements,

Conditional statements,

Unconditional statements.

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FORMATION OF FUZZY RULESAssignment Statements

Conditional Statements

Unconditional Statements

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DECOMPOSITION OF FUZZY RULES

A compound rule is a collection of several simple

rules combined together.

Multiple conjunctive antecedent,

Multiple disjunctive antecedent,

Conditional statements (with ELSE and

UNLESS).

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DECOMPOSITION OF FUZZY RULESMultiple Conjunctive

Antecedants

Conditional Statements ( With Else and Unless)

Multiple disjunctive

antecedent

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AGGREGATION OF FUZZY RULES

Aggregation of rules is the process ofobtaining the overall consequents from the individual consequents provided by each rule.

Conjunctive system of rules.

Disjunctive system of rules.

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AGGREGATION OF FUZZY RULES

Disjunctive system of rules

Conjunctive system of rules

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FUZZY RULE - EXAMPLE

Rule 1: If height is short then weight is light.

Rule 2: If height is medium then weight is medium.

Rule 3: If height is tall then weight is heavy.

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FUZZY RULE - EXAMPLE

Problem: Given(a) membership functions for short, medium-height, tall, light, medium-weight and heavy;(b) The three fuzzy rules;(c) the fact that John’s height is 6’1” estimate John’s weight.

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Solution:

(1) From John’s height we know that

John is short (degree 0.3)

John is of medium height (degree 0.6).

John is tall (degree 0.2).

(2) Each rule produces a fuzzy set as output by

truncating the consequent membership function

at the value of the antecedent membership.

FUZZY RULE - EXAMPLE

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FUZZY RULE - EXAMPLE

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FUZZY RULE - EXAMPLE

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FUZZY RULE - EXAMPLE

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FUZZY RULE - EXAMPLE The cumulative fuzzy output is obtained by OR-

ing the output from each rule.

Cumulative fuzzy output (weight at 6’1”).

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FUZZY RULE - EXAMPLE

1. De-fuzzify to obtain a numerical

estimate of the output.

2. Choose the middle of the range where

the truth value is maximum.

3. John’s weight = 80 Kg.

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FUZZY REASONING

There exist four modes of fuzzy

approximate reasoning, which include:

1. Categorical reasoning,

2. Qualitative reasoning,

3. Syllogistic reasoning,

4. Dispositional reasoning.

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REASONING WITH FUZZY RULES

In classical systems, rules with true antecedents fire.

In fuzzy systems, truth (i.e., membership in some class) is relative, so all rules fire (to some extent).

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REASONING WITH FUZZY RULES

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SINGLE RULE WITH SINGLE ANTECEDANT

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SINGLE RULE WITH MULTIPLE ANTECEDANTS

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MULTIPLE ANTECEDANTS

IF x is A AND y is B THEN z is CIF x is A OR y is B THEN z is C

Use unification (OR) or intersection (AND) operations to calculate a membership value for the whole antecedent.

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MULTIPLE RULE WITH MULTIPLE ANTECEDANTS

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MULTIPLE CONSEQUENTS

IF x is A THEN y is B AND z is C

Each consequent is affected equally by the membership in the antecedent class(es).

E.g., IF x is tall THEN x is heavy AND x has large feet.

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FUZZY INFERENCE SYSTEMS (FIS)

Fuzzy rule based systems, fuzzy models, and fuzzy expert systems are also known as fuzzy inference systems.

The key unit of a fuzzy logic system is FIS.

The primary work of this system is decision-making.

FIS uses “IF...THEN” rules along with connectors “OR” or “AND” for making necessary decision rules.

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The input to FIS may be fuzzy or crisp, but the

output from FIS is always a fuzzy set.

When FIS is used as a controller, it is

necessary to have crisp output.

Hence, there should be a defuzzification unit

for converting fuzzy variables into crisp variables

along FIS.

FUZZY INFERENCE SYSTEMS

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BLOCK DIAGRAM OF FIS

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TYPES OF FIS

There are two types of Fuzzy Inference

Systems:

Mamdani FIS(1975)

Sugeno FIS(1985)

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MAMDANI FUZZY INFERENCE SYSTEMS

(FIS) Fuzzify input variables:Determine membership values.

Evaluate rules:Based on membership values of (composite)

antecedents.

Aggregate rule outputs:Unify all membership values for the output from

all rules.

Defuzzify the output:COG: Center of gravity (approx. by summation).

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SUGENO FUZZY INFERENCE SYSTEMS

(FIS)The main steps of the fuzzy inference process namely,

1.fuzzifying the inputs and

2.applying the fuzzy operator

are exactly the same as in MAMDANI FIS.

The main difference between Mamdani’s and Sugeno’s methods

is that Sugeno output membership functions are either linear or

constant.

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SUGENO FIS

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FUZZY EXPERT SYSTEMS

An expert system contains three major blocks:

Knowledge base that contains the knowledge specific to the domain of application.

Inference engine that uses the knowledge in the knowledge base for performing suitable reasoning for user’s queries.

User interface that provides a smooth communication between the user and the system.

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BLOCK DIAGRAM OF FUZZY EXPERT SYSTEMS

Examples of Fuzzy Expert System include Z-II, MILORD.

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SUMMARY

Advantages of fuzzy logic

• Allows the use of vague linguistic terms in the

rules.

Disadvantages of fuzzy logic

• Difficult to estimate membership function

• There are many ways of interpreting fuzzy

rules, combining the outputs of several fuzzy

rules and de-fuzzifying the output.

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FUZZY DECISION MAKING

Decision making is a very important social,

economical and scientific endeavor.

The decision-making process involves

three steps:

1. Determining the set of alternatives,

2. Evaluating alternatives,

3. Comparison between alternatives.

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FUZZY DECISION MAKING

Certain fuzzy decision-making process are as

follows:

Individual decision making,

Multiperson decision making,

Multiobjective decision making,

Multiattribute decision making,

Fuzzy Bayesian decision making.

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INDIVIDUAL DECISION MAKING

April 2007 141

MULTIPERSON DECISION MAKINGIn multiperson decision making, the decision makers have

access to different information upon which to base their decision.

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MULTIOBJECTIVE DECISION MAKING

In making a decision when there are several objectives to be realized, then the decision making is called multiobjective decision making.

Many decision processes may be based on single objectives such as cost minimization, time consumption, profit maximization and so on.

The main issues in multiobjective decision making are:

To acquire proper information related to the satisfaction of the objectives by various alternatives. To weigh the relative importance of each objective.

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MULTIATTRIBUTE DECISION MAKING

The evaluation of alternatives can be carried out

based on several attributes of the object called

multiattribute decision making.

The attributes may be classified into numerical

data, linguistic data and qualitative data.

April 2007 144

The problem of decision-making structure in multiattributes

deals with determination of an evaluation structure for the

multiattribute decision making from the available

multiattribute data xi (i = 1 to n) and alternative evaluations

y shown in the table.

MULTIATTRIBUTE DECISION MAKING

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FUZZY BAYESIAN DECISION MAKING

Probability theory. Bayesian inference:

• Use probability theory and information about independence.• Reason diagnostically [from evidence (effects) to conclusions (causes)] or casually (from causes to effects).

Bayesian networks:• Compact representation of probability distribution over a

set of propositional random variables.• Take advantage of independence relationships.

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FUZZY BAYESIAN DECISION MAKING

April 2007 147

FUZZY BAYESIAN DECISION MAKING

Expected Utility

April 2007 148

FUZZY BAYESIAN DECISION MAKING

April 2007 149

SUMMARY

The chapter has discussed the various decision-making process.

In many decision-making situations, the goals, constraints and consequences of the defined alternatives are known imprecisely, which is due to ambiguity and vagueness.

Methods for addressing this form of imprecision are important for dealing with many of the uncertainties we deal within human systems.